As an Ahmadi myself, allow me to present our belief. This philosophy is taught to us by Mirza Ghulam Ahmad (as). I have quoted his works where he expounds upon this belief himself after a brief explanation here.
The English translation commonly used by Ahmadis (that of Maulvi Sher Ali) of verse 33:40 is the following:
Muhammad is not the father of any of your men, but he is the Messenger of Allah and the Seal of the Prophets; and Allah has full knowledge of all things. (33:40)
So خاتم النبيين is translated as Seal of the Prophets. This is the main translation that we use when discussing this topic.
However, Ahmadis still also take it to mean Last of the Prophets in the sense that he is the final law-bearing prophet and he brought the last shariah, and that no prophet will ever replace him.
Verses of the Quran used by us Ahmadis to demonstrate this are the following:
" O children of Adam, if there come to you messengers from among you relating to my verses, then whoever fears allah and reforms - there will be no fear concerning them, nor will they grieve." (7:35)
"And whoso obeys Allah and this messenger of his shall be among those who Allah has bestowed his blessings, namely, the Prophets, the Truthful, the Martyrs, and the righteous. And excellent companions are these." (4:70)
When we look at the Ahadith, this point becomes even more expounded and the truth becomes apparant.
First of all, Ahmadis acknoledge the following hadith (and those similar to it).
The Prophet (ﷺ) said to 'Ali: "You are to me in the position that Harun was to Musa, except that there is no Prophet after me." (Tirmidhi 3730 Book 49, Hadith 127)
However, the problem comes with the advent of Jesus (as). Will he not be a prophet? Hadith very clearly say that he will be:
"....Allah's Apostle, Jesus, and his companions would supplicate Allah, Who would send to them insects (which would attack their necks) and in the morning they would perish like one single person. Allah's Apostle, Jesus, and his companions would then come down to the earth and they would not find in the earth as much space as a single span which is not filled with their putrefaction and stench. Allah's Apostle, Jesus, and his companions would then again beseech Allah..." (Sahih Muslim 2937a Book 54, Hadith 134)
In this hadith the phrase نبي الله عيسى comes up multiple times. Sunnah.com translated it as "Allah's apostle, Jesus." A more fitting translation of course would be "Allah's prophet, Jesus."
We also see.
"The Prophet (ﷺ) said: There is no prophet between me and him, that is, Jesus (ﷺ). He will descent (to the earth)...." (Sunan Abi Dawud 4324 Book 39, Hadith 34)
It should be noted that Ahmadi Muslims believe in the death of Jesus (as) as expounded by the Quran and Ahadith. A detailed list of proofs I have included here if one wishes to see: https://docs.google.com/document/d/1hiqgKcEo64B70XPkBxMNYXaYTFl6T47qV3XK1-MrtvY/edit?usp=sharing
However, to keep it brief the problem is that if Jesus (as) of old comes back he will violate the finality of Prophethood. The argument that he was a prophet before is meaningless since this hadith rejects the coming of any prophet after the Holy Prophet (sa) (لا نبي بعدي). In fact, if he is granted prophethood before, that is the real moment when the Holy Prophet's (sa) status as خاتم النبيين is violated.
Hadith do not reject the general concept that prophets can come:
"I heard Allah's Messenger (ﷺ) saying, "Nothing is left of the prophetism except Al-Mubashshirat." They asked, "What are Al-Mubashshirat?" He replied, "The true good dreams (that conveys glad tidings)." (Bukhari 6990 Book 91, Hadith 9)
"Allah's Messenger (ﷺ) said, "A good dream of a righteous man is one of forty-six parts of prophetism." (Bukhari 6983 Book 91, Hadith 2)
How do Ahmadis reconcile these Ahadith with خاتم النبيين and لا نبي بعدي?
Well clearly prophethood is still open for the ummah. However, it is not true prophethood but partial prophethood. And this is the prophethood that the coming Messiah of the Muslims would have. He would have the title, but he wouldn't be the true prophet in essence because his prophethood is nothing but a reflection of the prophethood of Muhammad (sa). Anything he gained is in obedience to him (sa).
Jesus (as) of old did not gain his prophethood through obedience of the Holy Prophet (sa) but it was granted independently by God. So his coming is outside the realm of خاتم النبيين. As Ahmadis like to say it, Jesus (as) of Bani Israil does not have the "Seal" of Muhammad (sa).
This topic, naturally, is tackled by Mirza Ghulam Ahmad (as) himself very thoroughly in his books. While it would be a lot to quote everywhere, I will mention a very concise explanation in Arabic of this concept in his urdu book Tawzeeh-e-Maram. The English translation (Elucidation of Objectives) has this Arabic passage and its translation on pages 17-18:
https://www.alislam.org/library/books/elucidation/Elucidation-of-Objectives.pdf
و السلام على من اتبع الهدى